How to bind the channel of rotation of the object to the position channel of another object?
This can rather easily be achieved by using simple drivers. 
I'd recommend watching this video series. The first video explains pretty good how it works. 

Here is a quick recording i did, showing how to do what you're asking.

Steps:


Select Empty, on transforms add a driver on Z Location.
In the input variables select the cube as input.
Change the type to X Rotation. Your Empty should now move along it's Z-axis when the cube is rotating around its X-axis
If you want to inverse the input, you can simply change the Expression from "var + 1" to "var * -1". (Multiplying something by negative 1 simply inverts its value)


Edit: And if you want to do it with a constraint, this setup seems to work for me:

Gifycat showing it.



Don't know too much about how the Transformation constraint works, so can't really explain well why this work but it works. If you want to invert the axis just swap around the 90 and -90 on X rotation.